929 research outputs found
Lower Bounds on Time-Space Trade-Offs for Approximate Near Neighbors
We show tight lower bounds for the entire trade-off between space and query
time for the Approximate Near Neighbor search problem. Our lower bounds hold in
a restricted model of computation, which captures all hashing-based approaches.
In articular, our lower bound matches the upper bound recently shown in
[Laarhoven 2015] for the random instance on a Euclidean sphere (which we show
in fact extends to the entire space using the techniques from
[Andoni, Razenshteyn 2015]).
We also show tight, unconditional cell-probe lower bounds for one and two
probes, improving upon the best known bounds from [Panigrahy, Talwar, Wieder
2010]. In particular, this is the first space lower bound (for any static data
structure) for two probes which is not polynomially smaller than for one probe.
To show the result for two probes, we establish and exploit a connection to
locally-decodable codes.Comment: 47 pages, 2 figures; v2: substantially revised introduction, lots of
small corrections; subsumed by arXiv:1608.03580 [cs.DS] (along with
arXiv:1511.07527 [cs.DS]
Spiral Tessellation on the Sphere
In this paper we describe a tessellation of the
unit sphere in the
3-dimensional space realized using a spiral
joining the north and the south poles. This tiling yields to a
one dimensional labeling of the tiles covering the whole sphere
and to a
1-dimensional natural ordering on the set of tiles of
the tessellation. The correspondence between a point on the
sphere and the tile containing it is derived as an analytical
function, allowing the direct computation of the tile. This
tessellation exhibits some intrinsic features useful for general
applications: absence of singular points and efficient tiles
computation. Moreover, this tessellation can be parametrized
to obtain additional features especially useful for spherical
coordinate indexing: tiles with equal area and good shape
uniformity of tiles. An application to spherical indexing of a
database is presented, it shows an assessment of our spiral tiling
for practical use
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